ﻻ يوجد ملخص باللغة العربية
We aim at the development and analysis of the numerical schemes for approximately solving the backward diffusion-wave problem, which involves a fractional derivative in time with order $alphain(1,2)$. From terminal observations at two time levels, i.e., $u(T_1)$ and $u(T_2)$, we simultaneously recover two initial data $u(0)$ and $u_t(0)$ and hence the solution $u(t)$ for all $t > 0$. First of all, existence, uniqueness and Lipschitz stability of the backward diffusion-wave problem were established under some conditions about $T_1$ and $T_2$. Moreover, for noisy data, we propose a quasi-boundary value scheme to regularize the mildly ill-posed problem, and show the convergence of the regularized solution. Next, to numerically solve the regularized problem, a fully discrete scheme is proposed by applying finite element method in space and convolution quadrature in time. We establish error bounds of the discrete solution in both cases of smooth and nonsmooth data. The error estimate is very useful in practice since it indicates the way to choose discretization parameters and regularization parameter, according to the noise level. The theoretical results are supported by numerical experiments.
A thorough backward stability analysis of Hotellings deflation, an explicit external deflation procedure through low-rank updates for computing many eigenpairs of a symmetric matrix, is presented. Computable upper bounds of the loss of the orthogonal
Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which does not satisfy detailed balance. For the corresponding Fokker-Planck equation on a closed manifo
In this work, we study the numerical approximation of a class of singular fully coupled forward backward stochastic differential equations. These equations have a degenerate forward component and non-smooth terminal condition. They are used, for exam
This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as speci
In this paper we consider the numerical solution of Boussinesq-Peregrine type systems by the application of the Galerkin finite element method. The structure of the Boussinesq systems is explained and certain alternative nonlinear and dispersive term